Investors choosing a portfolio strategy, in order to secure a pension at a future date for
example, are faced with many uncertainties. One major uncertainty is the amount by
which their pension fund will be supplemented by personal savings from a variety of sources
such as life insurance contracts, bequests, or property sales. Over long periods of time these
uncertainties are likely to be large and difficult to hedge, and hence may have a significant
effect on the dynamic portfolio strategy. Drawing on the results of previous literature on the
reaction of investors to non-unhedgeable background risk, and on the theory of stochastic
dynamic programming, this article derives optimal strategies for investors maximising the
expected utility of terminal wealth, where this wealth consists of the value of a pension
fund plus accumulated personal savings. Numerical results, assuming that the market
portfolio and the expectation of personal savings follow (possibly) correlated geometric
Brownian motions, are derived to illustrate the effects of the size and uncertainty of the
personal savings, as well as the effect of the resolution of the uncertainty in them over
time. The computation uses a new technique for implementing the stochastic dynamic
programming. This involves a binomial approximation, in two dimensions, which ensures
that the computations are feasible for relatively long-term problems.